Strong Gelfand subgroups of $F\wr S_n$

TL;DR

This paper characterizes strong Gelfand subgroups of wreath products, providing classification results for multiplicity-free subgroups of symmetric and hyperoctahedral groups, with implications for representation theory.

Contribution

It offers a complete classification of multiplicity-free subgroups in wreath products and hyperoctahedral groups, including new decomposition formulas.

Findings

Wreath product $F\wr S_\lambda$ is multiplicity-free iff $\lambda$ has at most two parts.

Classified all multiplicity-free subgroups of hyperoctahedral groups.

Derived new decomposition formulas for induced representations.

Abstract

The multiplicity-free subgroups (strong Gelfand subgroups) of wreath products are investigated. Various useful reduction arguments are presented. In particular, we show that for every finite group $F$, the wreath product $F\wr S_\lambda$, where $S_\lambda$ is a Young subgroup, is multiplicity-free if and only if $\lambda$ is a partition with at most two parts, the second part being 0,1, or 2. Furthermore, we classify all multiplicity-free subgroups of hyperoctahedral groups. Along the way, we derive various decomposition formulas for the induced representations from some special subgroups of hyperoctahedral groups.